1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
|
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head><title>bnewbold thesis</title></head>
<body style="margin: 25px; font-family: helvetica;">
<h1 style="border-bottom: 2px solid;">
Journal: Feb 16, 2009</h1>
<i>Bryan Newbold, <a href="mailto:bnewbold@mit.edu">bnewbold@mit.edu</a></i><br />
<i><a href="http://web.mit.edu/bnewbold/thesis/">
http://web.mit.edu/bnewbold/thesis/</a></i>
<br /><p />
<!-- ================================================================ -->
<!-- ================================================================ -->
This past week we've been covering generic operators in 6.945, which is great
because it's a set of questions I'm just getting around to. What are the
important data-types/-structures/objects in general relativity, what are the
important operators and procedures, and what are their arities and
applicabilities?
Here are some potential object types (I'm using "object" in a general sense,
not with the usual CS implications):
<ul>
<li />manifold (arbitrary dimension)
<li />vector
<li />tensor
<li />vector and tensor fields
<li />metric (a rank-2 tensor)
<li />1-form (with implied metric)
<li />chart, atlas
<li />coordinate system
<li />parameterized curve
<li />coordinate basis
<li />intervals
<li />Hamiltonian, Lagrangian
<li />geodesic
</ul>
Some basic operators:
<ul>
<li />norm/inner-product
<li />contraction
<li />tensor raising/lowering (w.r.t. a metric)
<li />differential form
<li />symmetrize/anti-symmetrize
</ul>
Some tests:
<ul>
<li />Lorentzian? (of transformations)
<li />Poincare? (Lorentzian plus translation)
<li />timelike?, spacelike?, lightlike? (of intervals and curves)
<li />zero-curvature? (of manifolds)
<li />rank? (of tensors)
<li />dimension? (of manifolds)
</ul>
Some more specific operators:
<ul>
<li />Riemann curvature
</ul>
And some tools:
<ul>
<li />evolve
<li />find-geodesic
<li />find-interval
</ul>
<hr />
Turns out I wasn't using the latest version of scmutils and that's why a bunch
of calculus functionality was missing... whoops!
<hr />
Time is tick ticking away but I feel that things are starting to fall together,
doing problem sets for my GR class is super valuable, I wish i'd done more
practice exercises before to test myself. But overall I feel pretty
mathematically prepared.
<br />
Over the weekend I went back and tried to review some topics from topology like
Baire dimension theory, embedding, and the fundamental group. I never really
covered these topics when I studies topology before, so I wasn't able to gleam
any insight.
<br /><br />
<a href="05feb2009.html"><i>(previous entry)</i></a> -
<a href="19feb2009.html"><i>(next entry)</i></a>
<!-- ================================================================ -->
<!-- ================================================================ -->
</body>
</html>
|
